# Bayesian Data Analysis

Many fields of science are transitioning from null hypothesis significance testing (NHST) to Bayesian data analysis. Bayesian analysis provides rich information about the relative credibilities of all candidate parameter values for any descriptive model of the data, without reference to *p* values. Bayesian analysis applies flexibly and seamlessly to complex hierarchical models and realistic data structures, including small samples, large samples, unbalanced designs, missing data, censored data, outliers, etc. Bayesian analysis software is flexible and can be used for a wide variety of data-analytic models. This course shows you how to do Bayesian data analysis, hands on, with free software called R and JAGS. The course will use new programs and examples.

This course is closely modeled on the very successful series of workshops given by Prof. John Kruschke. We will be using his software, and I strongly recommend his book (see below) and his blog, doingbayesiandataanalysis.blogspot.com

**Course Objectives: **You will learn

- the rich information provided by Bayesian analysis and how it differs from traditional (Frequentist) statistical analysis
- the concepts of Bayesian reasoning along with the easy math and intuitions for Bayes’ rule
- the concepts and hands-on use of modern algorithms (“Markov chain Monte Carlo”) that achieve Bayesian analysis for realistic applications
- how to use the free software R and JAGS for Bayesian analysis, with many programs created by the instructor, readily useable and adaptable for your research applications
- an extensive array of applications, including comparison of two groups, ANOVA- like designs, linear regression, logistic regression, ordinal regression, etc. Also numerous variations for robustness to outliers, non-normally distributed noise, heterogenous variances, censored data, non-linear trends, auto-regressive models, etc. See more details in the list of topics, below.

**Course Audience
**The intended audience is PhD students, faculty, and other researchers, from all disciplines, who want a ground-floor introduction to doing Bayesian data analysis.

**Course Prerequisites
**No specific mathematical expertise is presumed. In particular, no matrix algebra is used in the course. Some previous familiarity with statistical methods such as a t-test or linear regression can be helpful, as is some previous experience with programming in any computer language, but these are not critical.

**Course Topics **(Exact content, ordering, and durations may change.)

*Day 1:*

- Overview / Preview:
- Bayesian reasoning generally. (See this introductory chapter.)
- Robust Bayesian estimation of difference of means. Software: R, JAGS, etc.
- NHST t test: Perfidious p values and the con game of confidence intervals.

- Bayes’ rule, grid approximation, and R. Example: Estimating the bias of a coin.
- Markov Chain Monte Carlo and JAGS. Example: Estimating parameters of a normal distribution.
- HDI, ROPE, decision rules, and null values.

*Day 2: *

- Hierarchical models: Example of means at individual and group levels. Shrinkage.
- Examples with beta distributions: therapeutic touch, baseball, meta-analysis of extrasensory perception.
- The generalized linear model.
- Simple linear regression. Exponential regression. Sinusoidal regression, with autoregression component.
- How to modify a program in JAGS & rjags for a different model.
- Robust regression for accommodating outliers, for all the models above and below.
- Multiple linear regression.
- Logistic regression.
- Ordinal regression.
- Hierarchical regression models: Estimating regression parameters at multiple levels simultaneously.

*Day 3:*

- Hierarchical model for shrinkage of regression coefficients in multiple regression.
- Variable selection in multiple linear regression.
- Model comparison as hierarchical model. The Bayes factor. Doing it in JAGS.
- Two Bayesian ways to assess null values: Estimation vs model comparison.

*Day 4:*

- Bayesian hierarchical oneway “ANOVA”. Multiple comparisons and shrinkage.
- Example with unequal variances (“heteroscedasticity”).
- Bayesian hierarchical two way “ANOVA” with interaction. Interaction contrasts.
- Split plot design.
- Log-linear models and chi-square test.

*Day 5:*

- Power: Probability of achieving the goals of research. Applied to Bayesian estimation of two groups.
- Sequential testing.
- The goal of achieving precision, instead of rejecting/accepting a null value.
- How to report a Bayesian analysis.
- Advanced topics as time permits and audience interest suggests:
- Censored data in JAGS.
- Mixture of normals
- Other data distributions in JAGS using Bernoulli 1’s trick.
- Stan and Hamiltonian Monte Carlo

**Highly recommended textbook**

*Doing Bayesian Data Analysis, 2nd Edition: A Tutorial with R, JAGS, and Stan*. The book is a genuinely accessible, tutorial introduction to doing Bayesian data analysis. The software used in the course accompanies the book, and many topics in the course are based on the book. (The course uses the 2nd edition, not the 1st edition.) Further information about the book can be found at https://sites.google.com/site/doingbayesiandataanalysis/.

**Install software before arriving**

It is important to bring a notebook computer to the course, so you can run the programs and see how their output corresponds with the presentation material. Please install the software before arriving at the course. The software and programs are occasionally updated, so please check here a week before the course to be sure you have the most recent versions.

**For complete installation instructions, please go to ****https://sites.google.com/site/doingbayesiandataanalysis/software-installation**

**Examination**

Examination paper written at home (individual) 100%

**Examination content**

For students taking the course for credit, there are daily homework exercises, all due one week after the last day of class**.**

Students are encouraged to use whatever resources they can to successfully execute the homework exercises, but the ultimate execution and write‑up must be by their own hand and in their own words. An honor statement must accompany each submitted assignment.

Each day’s homework exercises will be computer‑based replications and extensions of examples in that day’s lecture. The exercises ask the student to reproduce the example, explain its meaning, and perform a specific extension or novel application.

**Examination relevant literature**

There is no additional literature needed for the homework exercises.